So, basically in one state, A is connected to X and B is connected to Y, but in the other state, A is connected to Y and B is connected to X.

I spent some time googling around and reading answers seem to be pointing me towards using a mux, but that would only switch one of the outputs. What would I use to get the other output to be routed to the opposite of what the mux selects?

Or...I'm sure there's another way to do what I'm trying to get at? (The beginner terms for googling this kind of thing (switch, cross, flip), aren't the best, as you can imagine.)

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\$\begingroup\$What kind of signal [or power] are you routing? Do you need a dry connection (kind of relay?) Or perhaps some digital logic will suffice?\$\endgroup\$
– Eugene Sh.Jul 31 '18 at 15:30

\$\begingroup\$You can buy analogue switches and multiplexers in all different sizes and shapes. Some four channel some 8 channel some single channel. Some with c/o contacts some with just n/o contacts. I use the word "contact" loosely of course.\$\endgroup\$
– Andy akaJul 31 '18 at 15:58

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\$\begingroup\$@user5265147: "... so bare with me" is an invitation to undress together. You mean "... so bear with me" as in "to bear one another's burdens".\$\endgroup\$
– TransistorJul 31 '18 at 16:30

3 Answers
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What you want is called a Bus Exchange Switch. As an example, the logic diagram from the Onsemi 7WB383 is shown below. It connects either straight through, swapped, or not at all, depending on the control inputs.

The switches are available in different logic families and voltages. Googling the term "bus exchange switch" will find you many possibilities.